Re: What is the relationship between velocity and the diameter of a wheel?

I want to know if there is a relationship between the velocity of an
object with wheels when put down a ramp and the diameter of the wheels of
the object. If there is a relationship, what is it?

Hi Morgan,

Intuitively, you'd sorta expect there to be some dependence of the velocity of a wheel on its diameter... and you'd be correct. There is a dependence but it sneaks in mainly via the moment of inertia of the object (which depends on the radius for regular objects). I'll derive the relationship for you and then point out a simplification that'll probably come in handy in most normal situations. I'm not sure if you've taken a physics course at your school yet but this derivation can get pretty involved. I'll try my best to keep the hairier aspects at bay but if one or two sneak in, I'd really recommend checking out some of the sources I list at the end (and, of course, please feel free to drop me a line for any clarification). Okay, on to the answer...

Before we get started, though, like all good scientists, let's make our language a little more precise so that we know exactly what we're talking about and avoid ambiguity. The velocity I suspect you're interested in is the translational velocity of the object; in other words, how fast it goes linearly as opposed to how fast it's rotating (i.e., it's angular velocity).

There are many ways to go about solving such a problem. Some are quite simple and straightforward while others, as you can well imagine, are highly involved and very complicated. For our present circumstances, let's try to keep things as simple as possible. As such, we'll allow for many idealizations and choose the easiest possible path to a solution. The "easiest possible path to a solution", for this case, turns out to be looking at the situation from a energy perspective (as opposed to a vector forces approach). If we imagine an ideal situation where we have a cylinder (instead of an object with wheels) rolling down a ramp without slipping (this is another important idealization) then the situation looks something like this (please note that the following image was obtained from The Interactive Textbook site which also has a great discussion of a similar problem from the vector point of view... give it a look when you have some free time):

When the cylinder is at the top of the ramp, it is stationary. Since it is completely still, the only energy it possesses is the energy of position, or potential energy. For a cylinder of mass m, this Potential Energy = mgh, where h is the height of the ramp and g is the gravitational acceleration. Now, as it rolls down the incline, it acquires both translational kinetic energy (KE_Tr = ¹/₂mv², where v is the translational velocity of the center of mass of the object) and rotational kinetic energy (KE_Rot = ¹/₂Iw², where I is the moment of inertia for the object and w is the angular velocity of the object.). Now Conservation of Energy tells us that total initial energy = total final energy so, for our cylinder, this statement translates into:

(1) E_i = E_f
(2) PE = KE_Tr + KE_Rot
(3) mgh = ¹/₂mv² + ¹/₂Iw²

Now we can bring in another idealization: since we assumed that our object was a perfect cylinder, we can use the moment of inertia of a cylinder (I_cyl=¹/₂mR², where R is the radius of the cylinder) in equation (3) above to get:

(4) mgh = ¹/₂mv² + ¹/₂(¹/₂mR²)w² = ¹/₂mv² + ¹/₄mR²w²

Since we also placed the constraint that the cylinder rolls without slipping, the angular velocity of the cylinder is related to the translational velocity of the cylinder by w = v/R so equation (4) becomes:

(5) mgh = ¹/₂mv² + ¹/₄mR²(^V/_R) ² = ¹/₂mv² + ¹/₄mv² = ³/₄mv²

The final velocity of the cylinder at the bottom of the incline becomes v = sqrt[⁴/₃gh]. So we see that, for a cylinder rolling down an incline, the radius cancels out but a factor of sqrt[4/3] comes in because the cylinder rotates as it comes down (if there was no rotation, then the factor would've been just the usual sqrt[2]). Now we can finally see the relation between the sizes of different objects (e.g., different diameters, different masses, etc.) and the final velocity they might have. Different objects of different sizes have different moments of inertia and since the moment of inertia comes into equation (3), it will change the factor from sqrt[4/3] to some other value. Thus, the factor by which the velocity differs for different objects depends on the composition of the body (the mass, how the mass is distributed, the radius, etc.).

As an example, if we had rolled a sphere (with a moment of inertia I_sph = ²/₅mR²) down the incline instead of our cylinder, then our final factor would've been sqrt[10/7] instead of sqrt[4/3]. Now, in our example, with its many idealizations and assumptions, things worked out rather neatly. But in a real example, things would get quite a bit more complicated and the dependence on the radius would be even more pronounced than just the implicit dependence through the factor of the moment of inertia we derived above. As a little more complicated example, if we'd considered a thick ring rolling down the incline instead of our cylinder, then the ring, with a moment of inertia I_ring = ¹/₂m(R₁² + R₂²), would have yielded a equation (5) that would've been considerably more complicated since the R's wouldn't have cancelled out as easily as they did in our simplified example with the cylinder.

I know this might have been a little more complicated than you expected but if you give it some time, you'll find that it's easily digested. As an aid, I'd definitely recommend looking at the site I mentioned earlier. In addition, scan through some of the other answers on the MadSci network. But the best resource, of course, is your local library. One particularly good book that I still use every so often is The Cartoon Guide to Physics. Although it doesn't discuss rotations in quite so much detail, it's a good way to get your mind around some of the basic physics that's involved. Besides this, there are some great introductory books by such authors as Beiser, Giancoli, etc. And, if you feel like getting your hands dirty, I'd recommend one of the best introductions to physics, The Feynman Lectures. Although Feynman's target audience was undergrads, I would still highly recommend giving it a shot. And, of course, if all else fails, please feel free to drop me a line at rickys@sethi.org if you'd like anything clarified further (or if you find any errors or murky areas in my analysis). I hope this helped a little!

Best regards,


Rick.